Advection-dispersion Equation Model Research Articles

Scale Dependence of Dispersion Coefficient for Solute Transport in Porous Media Using Image Analysis

Scale dependence of dispersion coefficient (D) in the advection-dispersion equation (ADE) for solute transport in porous media was investigated by a series of experiments using image analysis. A hexahedral plexiglass box sized 200×8×1.5 cm (L×W×H) was set and packed with glass beads as porous media. The solute transport under different conditions was simulated by changing the particle size of glass beads, flow rate, and detection scale using Bright Blue as tracer. The image analysis method was used to dynamically monitor and identify the spatiotemporal variation of solute concentration distribution. The results showed that image analysis can effectively monitor and identify the solute concentration in porous media, as indicated by an R2 value of 0.9890. There is an obvious linear relationship between hydraulic gradient (J) and velocity (v) in porous media under different experimental conditions. The ADE model is suitable for solute breakthrough curve (BTC) with good fitting accuracy, and can effectively reflect the concentration variation during solute transport. The key parameters controlling the solute transport were analyzed. D has abnormal diffusion (i.e., non-Fickian phenomenon) and scale dependence, and BTCs had a long tail, which becomes more obvious with the increase of flow rate, medium particle size, and transport scale.

Migration modelling of As(V) loaded by humic acid and nano iron oxide composite colloids affected by various environmental factors

• Adsorption of nano-iron oxide on arsenic diminishes with increasing humic acid, pH, and iron oxide. • Increasing ionic strength has an inhibitory effect on the arsenic migration. • The fractal model can accurately capture the arsenic migration in porous media. Nanoparticles have been applied to remediate heavy metal pollutants (e.g., arsenic) in groundwater and soil, where the migration mechanism of pollutants in this multi-component system has not been fully understood. To better understand how nano-iron oxide affects arsenic migration under multivariate influence, this study prepared arsenic loaded composite colloids using arsenic (As(V)), humic acid, and nano iron oxide (n-α-Fe 2 O 3 ). Then we explored the migration behavior of the composite colloids (HA-n-α-Fe 2 O 3 ) loaded As(V) in a quartz sand column affected by n-α-Fe 2 O 3 under various humic acid concentrations, pH values, ionic strengths, and ferric oxide contents. Two migration models, including a classical advection-dispersion equation (ADE) and a Hausdorff fractal advection-dispersion equation (HADE), were used to quantify the observed co-migration of arsenic conveyed by HA-n-α-Fe 2 O 3 colloids loaded As(V) in the column. Our analysis indicates that the adsorption capability of nano-iron oxide on arsenic decreases with pH, humic acid and iron oxide concentrations, thus accelerating arsenic movement. While, the increasement of ionic strength enhances the adsorption force of nano-iron oxide on arsenic and suppresses arsenic migration. Furthermore, simulation results suggest that the HADE model outperforms the traditional ADE model in characterizing arsenic migration, where the time derivative index is an indicator of anomalous diffusion.

Bayesian inversion of laboratory experiments of transport through limestone fractures

In this study, a novel experimental setup is proposed for which a column filled with glass beads and parallelepiped-shaped limestone beams is used to reconstruct a multiple fracture limestone media. The proposed setup produces asymmetric breakthrough curves (BTCs) that are consistent with the shape expected from the past field and lab-scale studies. Three transport experiments have been conducted under fast, medium, and slow flow velocity conditions.The research focuses on parameter and state estimation using Bayesian inference via Markov Chain Monte Carlo (MCMC) sampler, investigating the degree to which three models of transport through fractured media can reproduce the experimental results under the three flow conditions. The first transport model, named ADE, is based on the equivalent porous medium (EPM) approach and corresponds to the linear advection dispersion equation (ADE). The second model, named FOMIM (first-order mobile immobile), is based on the mobile/immobile approach and uses the dual porosity model with a linear first-order transfer between mobile and immobile regions. The third model, named NLMIM (non-linear mobile-immobile), uses a nonlinear transfer function between these two regions.The results of the three models show that almost all the unknown model input parameters can be well-estimated with narrow confidence intervals using the MCMC method. With respect to state estimation, the ADE model fails to reproduce correctly the tail of the BTCs observed under slow and medium flow conditions. The FOMIM model improves the tailing of the BTCs, but significant discrepancies remain between simulated and measured concentrations. The NLMIM model with velocity-dependent parameters is the only model that captures BTCs under all three conditions of slow, medium, and fast flow velocities.

Global fallout Pu isotopes, 137Cs and 237Np records in the sediments of Lake Xingkai and their response to environmental changes in the catchment

Plutonium (Pu) isotopes, neptunium-237 (237Np) and cesium-137 (137Cs) in the environment have emerged as important tools for promoting geochemical studies. To elucidate their source terms, deposition process, and the geochemical behavior, as well as their response to catchment environmental changes in lacustrine environment, 239, 240Pu, 137Cs and 237Np were determined for the first time in a sediment core collected from the Lake Xingkai. The 240Pu/239Pu atom ratios (0.163–0.178) in all sediment samples corresponded well with the typical global fallout value of ∼0.18, indicating that there were no other significant radioactive source inputs. The activity concentrations of 239+240Pu, 137Cs, and 237Np were 0.010–1.030 Bq kg−1, 6.4–30.1 Bq kg−1, and (0.020–2.195) × 10−3 Bq kg−1, respectively. They presented similar vertical distribution patterns, with a typical global fallout peak (15–16 cm) corresponding to the year of 1964. On the other hand, the 237Np/239Pu atom ratios exhibited a wide range (0.207–1.306), and showed a trend of increasing values with depth, implying different geochemical behaviors of these two elements. Using the advection-dispersion equation model (ADE), the obtained apparent dispersion coefficient of 237Np (0.58 ± 0.09 cm2 yr−1) was higher than that of 239+240Pu (0.38 ± 0.07 cm2 yr−1), indicating that Np has greater mobility than Pu. The total global fallout inventories of 239+240Pu, 137Cs, and 237Np were 145, 4592, and 0.291 Bq m−2, respectively. Approximately 46% of the total global fallout was derived from riverine input, indicating that severe soil erosion occurred in the catchment of Lake Xingkai in the past decades.

Co-transport and co-release of Eu(III) with bentonite colloids in saturated porous sand columns: Controlling factors and governing mechanisms

Accurate prediction of the colloid-driven transport of radionuclides in porous media is critical for the long-term safety assessment of radioactive waste disposal repository. However, the co-transport and corelease process of radionuclides with colloids have not been well documented, the intrinsic mechanisms for colloids-driven retention/transport of radionuclides are still pending for further discussion. Thus the controlling factors and governing mechanisms of co-transport and co-release behavior of Eu(III) with bentonite colloids (BC) were discussed and quantified by combining laboratory-scale column experiments, colloid filtration theory and advection dispersion equation model. The results showed that the role of colloids in facilitating or retarding the Eu(III) transport in porous media varied with cations concentration, pH, and humic acid (HA). The transport of Eu(III) was facilitated by the dispersed colloids under the low ionic strength and high pH conditions, while was impeded by the aggregated colloids cluster. The enhancement of Eu(III) transport was not monotonically risen with the increase of colloids concentration, the most optimized colloids concentration in facilitating Eu(III) transport was approximately 150 mg L−1. HA showed significant promotion on both Eu(III) and colloid transport because of not only its strong Eu(III) complexion ability but also the increased dispersion of HA-coated colloid particles. The HA and BC displayed a synergistic effect on Eu(III) transport, the co-transport occurred by forming the ternary BC-HA-Eu(III) hybrid. The transport patterns could be simulated well with a two-site model that used the advection dispersion equation by reflecting the blocking effect. The retarded Eu(III) on the stationary phase was released and remobilized by the introduction of colloids, or by a transient reduction in cation concentration. The findings are essential for predicting the geological fate and the migration risk of radionuclides in the repository environment.

Influence of Shear Displacement on Fluid Flow and Solute Transport in a 3D Rough Fracture

Abstract Fractured rocks in the subsurface are ubiquitous, and the dynamics of mass transfer in fractured rocks plays an important role in understanding the problem in engineering geology and environmental geology. In this study, the influence of shear displacement on fluid flow and solute transport in a 3D rough fracture was investigated. A 3D self-affine rough fracture was generated using the modified successive random addition (SRA) technology, and three sheared fractures with different shear displacements were constructed based on the mechanistic model. A direct numerical model based on the Navier-Stokes equation and the advection-diffusion equation was developed to solve the fluid flow and the solute transport. The results showed that shear displacement had a significant influence not only on the fluid flow but also on the solute transport. A global measure of the spatial variability of the flow velocity showed that the heterogeneity became weaker with decreasing shear displacement. All measured BTCs deviated from the Gaussian profile and exhibited the typical anomalous behaviors, such as the long tail and the early arrival. Although the best-fitted results of the advection-dispersion equation (ADE) model and mobile-immobile model (MIM) were generally consistent with those of the BTCs, the MIM was more capable than the ADE model for characterizing the shear-induced anomalous behavior of the BTCs. It was found that the mass exchange process between the immobile and mobile domains was enhanced in the sheared fractures while the fraction of the advection-dominant mobile domain decreased as the shear displacement increased. Furthermore, the deviation of the Taylor dispersion coefficient from the fitted dispersion coefficient by the ADE model and MIM in the sheared fractures was confirmed due to the influence of shear displacement.

Influence of Grain Size Transition on Flow and Solute Transport through 3D Layered Porous Media

Abstract Soils and other geologic porous media often have contrasting grain size layers associated with a grain size transition zone between layers. However, this transition zone is generally simplified to a plane of zero thickness for modeling assumption, and its influence has always been neglected in previous studies. In this study, an approach combining a deposition process and a random packing process was developed to generate 3D porous media without and with consideration of the transition zone. The direct numerical models for solving the flow and concentration fields were implemented to investigate the influence of the grain size transition on flow and solute transport. Our results showed that although the transition zone occupied 13.6% of the entire layered porous medium, it had little influence on the distribution of flow velocity at the scale of the entire layered porous medium. However, the transition zone had a significant influence on the local flow field, which was associated with the increased spatial variability of velocity and the varied distribution of flow velocity. This varied local flow field could increase the solute residence time and delay the breakthrough time for solute transport. Although using both the advection-dispersion equation (ADE) and the mobile and immobile (MIM) models to fit the breakthrough curves (BTCs) for solute transport through layered porous media resulted in trivial errors, the ADE model failed to capture the influence induced by the local flow field, especially the influence of the transition zone. In contrast, the MIM model was shown to be able to capture the influence of the transition on solute transport. It was found that the mass transfer rate α, a parameter of the MIM model, was significantly improved by the presence of the transition zone, while it decreased as the transition zone fraction increased. Our study emphasized that the transition zone can vary the local flow field at the pore scale, while it has little influence on the hydraulic properties (e.g., hydraulic conductivity) of the macroscale flow field. However, the local flow field varied by the transition zone has a significant influence on solute transport.

Experimental Research on the Thresholds of Scale-Dependent Dispersivity of Solute Transport

Abstract The conventional advection-dispersion equation (ADE) has been widely used to describe the solute transport in porous media. However, it cannot interpret the phenomena of the early arrival and long tailing in breakthrough curves (BTCs). In this study, we aim to experimentally investigate the behaviors of the solute transport in both homogeneous and heterogeneous porous media. The linear-asymptotic model (LAF solution) with scale-dependent dispersivity was used to fit the BTCs, which was compared with the results of the ADE model and the conventional truncated power-law (TPL) model. Results indicate that (1) the LAF model with linear scale-dependent dispersivity could better capture the evolution of BTCs than the ADE model; (2) dispersivity initially increases linearly with the travel distance and is stable at some limited value over a large distance, and a threshold value of the travel distance is provided to reflect the constant dispersivity; and (3) compared with the TPL model, both the LAF and ADE models can capture the behavior of solute transport as a whole. For fitting the early arrival, the LAF model is less than the TPL; however, the LAF model is more concise in mathematics and its application will be studied in the future.

Fractional Derivative Modeling on Solute Non-Fickian Transport in a Single Vertical Fracture

Solute transport in a single vertical fracture (SVF) cannot be reliably described by the classical advection-dispersion equation (ADE) model, due to the heterogeneity nature of fracture. This study conducted a group of experiments to investigate chloride ion transport in the SVFs under different rough-walled conditions, and then applied a time fractional advection-dispersion equation (F-ADE) model to offer an accurate description. A comparison between F-ADE model and a classical ADE model in describing experimental data, was also carried out. Results show that the FADE model is better than the ADE model in describing the breakthrough curve and heavy-tail phenomenon of solute transport in the fracture. Especially in the experiments with lower flow rate and higher roughness fracture, the FADE model can offer an accurate description, indicating that it is a promising tool for characterizing solute transport heterogeneous vertical fracture.

Experimental and modeling evidence of kilometer-scale anomalous tracer transport in an alpine karst aquifer

Karst aquifers are important drinking water resources, but highly vulnerable to contamination. Contaminants can be transported rapidly through a network of fractures and conduits, with only limited sorption or degradation, which usually leads to a fast and strong response at karst springs. During migration, contaminants can also enter less mobile zones, such as pools or water in intra-karstic sediments, or advance from conduits into the adjacent fractured rock matrix. As contaminant concentrations in the main flow path(s) decrease, contaminants may migrate back into the main flow path and reach the karst springs at low (but significant) concentrations over a long time span. This is the conventional interpretation for the oft-observed steep rising limb and the long-tailed falling limb of tracer breakthrough curves in karst systems. Here, field measurements are examined from an alpine karst system in Austria where a series of distinctive, long-tailed breakthrough curves (BTCs) of conservative tracers were observed over distances up to 7400 m. Recognizing that the conventional advection-dispersion equation (ADE) cannot usually quantify such behavior, two other modeling approaches are considered, namely the two-region non-equilibrium (2RNE) model, which explicitly includes mobile and immobile zones, and a continuous time random walk (CTRW) model, which is based on a physically-based, probabilistic approach that describes anomalous (or non-Fickian) transport behavior characteristic of heterogeneous systems such as karst. In most cases, the ADE and 2RNE models do not quantify the low concentrations at longer travel times. The CTRW, in contrast, accounts for the long-tailed breakthrough behavior found in this karst system.

Fundamental Study on Transport Model for Radionuclides under Unsaturated Condition around Near-Surface Underground

ABSTRACTThe low-level nuclear wastes such as decontamination waste from Fukushima are disposed in near-surface underground, where the intermittent recharge of rain and groundwater causes spatial distribution of water content. Therefore, pores of soils are not filled with water, that is, an unsaturated zone will be formed. In such a condition, since the water flow path are detoured by clogged gas in pores of soil in the unsaturated zone, the migration path of radionuclide would be different from the saturated zone. So far, the one-dimensional advection-dispersion equation (ADE) model has been widely used in order to explain experimental results under an unsaturated condition. However, the detouring of local flow-paths remarkably affects the mass transfer. The one-dimensional ADE evaluates such a detouring effect by using Peclet number and retardation coefficient as fitting parameters. In other words, the one-dimensional ADE model is difficult to explain mass transfer under an unsaturated condition. Therefore, the purpose of this study is explaining such complicated transport of radionuclides using a multi-path model based on phenomena in underground. The proposed multi-path model considering both water saturation and permeability distributions showed good agreement with the experimental data under an unsaturated condition.

An investigation on the fractional derivative model in characterizing sodium chloride transport in a single fracture⋆

Contaminant transport in a single fracture cannot be reliably captured by the classical advection-dispersion equation model, due to the heterogeneity nature of the fracture. This study applied a time fractional advection-dispersion equation (ADE) model to quantify chloride ion transport in a single fracture observed in the lab. We further explored the solute transport characteristics by comparing them with the simulation results using a classical ADE model and a space-time fractional ADE model. Comparison results show that the time fractional derivative model is better than the classical ADE model in describing the blocking process of solute transport in the fracture (the tailing phenomenon of the breakthrough curve or BTC). The space-time fractional ADE model is not needed to describe the observed BTC, because only time evolution of solute concentration is considered in this case. Experimental analysis also indicates that the order of the time fractional derivative increases with the fracture aperture, and sub-diffusion is enhanced by a slower flow velocity.

Effects of vermiculite, nanoclay and zeolite on ammonium transport through saturated sandy loam soil: Column experiments and modeling approaches

Economic and nature-friendly amendments are needed to increase fertilizer efficiency and crop yield for sustainable agriculture in coarse-textured soil. In this investigation, the effects of different application rates (2%, 4% and 8%) of vermiculite, nanoclay and zeolite amendments on adsorption and transport of ammonium (NH4-N) in sandy loam soil were studied using batch and column experiments. The results of batch experiment indicated that the maximum distribution coefficient (Kd) was obtained in soil amended with zeolite 8%, while minimum Kd was achieved in the control soil. Column experiments revealed that the application of inorganic amendments decreased the leaching of NH4-N from 89.4% in control soil to 77.6%, 70.2%, 52.5%, 85.3%, 60.8%, 38.5%, 12.6% and 2.1% in soil columns amended with vermiculite 2%, 4%, 8%, nanoclay 2%, 4% and zeolite 2%, 4%, 8%, respectively. On the basis of the application rate of inorganic amendments required to retain 40% of injected NH4-N, the effectiveness of zeolite is about 2 and 4 times that of nanoclay and vermiculite, respectively. The HYDRUS-1D program was used to simulate the leaching of NH4-N using the advection–dispersion equation model (ADE) along with batch and fitted Kd. The simulation results indicated that the ADE model with fitted Kd described the behavior of NH4-N transport more suitable than the ADE model using batch Kd (root-mean-square error, RMSE < 0.06 and RMSE < 0.13, respectively). Results of our study indicated that amending sandy loam soil with inorganic amendments can be an effective technique for decreasing NH4-N transport and improving the efficiency of N fertilizer in sustainable agriculture systems.

Revisit of advection-dispersion equation model with velocity-dependent dispersion in capturing tracer dynamics in single empty fractures

An accurate quantification of the contaminant transport through fractured media is critical for dealing with water-quality related scientific and engineering issues, where the dispersion coefficient is an important and elusive parameter for the solute transport modeling. Many previous studies show that the dispersion coefficient (D) in the standard advection-dispersion equation (ADE) model can be approximated by D = αvλ (where α is the dispersivity), a formula to be revisited systematically in this study by laboratory experiments and model analysis. First, a series of tracer transport experiments in single empty fractures are conducted in cases of different hydraulic gradients. Second, the tracer breakthrough curves are determined by simulations based on the ADE model, to obtain the dispersion coefficients corresponding to various fracture roughnesses and flow velocities. A varying trend of λ is analyzed under different flow conditions. Results show that although the standard ADE model cannot be used to characterize the late-time tailing of the tracer BTCs, likely due to the solute retention, this simple model can simulate most of the solute mass dynamics moving through fractures and may therefore provide information for estimating the dispersion in parsimonious models appropriate for the non-Fickian transport. The following three conclusions are drawn: (1) the peak of the breakthrough curves comes earlier with increasing the roughness, according to the ADE simulation, (2) the value of λ generally decreases as the relative roughness of the fracture increases, (3) the value of λ is approximately equal to 2.0 when the dispersion is dominated by the molecular diffusion in the smooth fracture.

Non-invasive image processing method to map the spatiotemporal evolution of solute concentration in two-dimensional porous media

A color visualization-based image processing method is developed in this paper to quantify the concentration evolution of the Brilliant Blue FCF transport through a two-dimensional homogeneous porous medium. A series of images are recorded at known time intervals, then the spatial distribution is estimated using a calibration curve, linking the gray pixel value to the solute concentration. Using a multi-dimensional concentration distribution map extraction technique the longitudinal and transverse concentration distributions could be observed with the physical model. The image-processed concentrations are then compared directly with the measured concentrations sampled at the outlet end. The tracer breakthrough curves sampled at multiple points along the central line of the medium are also compared with the solutions from the standard advection–dispersion equation model. It is shown that the non-invasive image processing method may be used to map the spatiotemporal evolution of a solute’s concentration without disturbing the flow or the transport dynamics, although the measured solute breakthrough curves feature some non-Fickian dynamics that cannot be efficiently captured by the standard transport model.

Transition from non-Fickian to Fickian longitudinal transport through 3-D rough fractures: Scale-(in)sensitivity and roughness dependence

Understanding transport in rough fractures from non-Fickian to Fickian regimes and the prediction of non-Fickian transport is critical for the development of new transport theories and many practical applications. Through computational experiments that fall within the macrodispersion regime, we first simulated and analyzed solute transport through synthetic rough fractures with stationary geometrical properties (i.e., fracture roughness σb/〈b〉 and correlation length λ, where b refers to aperture with its standard deviation σb and arithmetic mean 〈b〉) across increasing fracture longitudinal transport domain length L, with L/λ ranging from 2.5 to 50. The results were used to determine how solute transport behavior evolves with increasing scale in the longitudinal direction. Moreover, a set of correlated fractures with aperture fields following normal and log-normal distributions was created to further identify and quantify the dependence of non-Fickian transport on roughness. We found that although persistent intermittent velocity structures were present, the breakthrough curves (BTCs) and residence time distributions showed diminishing early arrival and tailing, features of non-Fickian transport, with increasing longitudinal L/λ, ultimately converging to a Fickian transport regime given σb/〈b〉 remained constant. Inverse analysis of the experimental BTCs with the advection-dispersion equation (ADE) model showed that the dispersion coefficient (D) was non-trivially scale-dependent. Simulation results for rough fractures with varying σb/〈b〉 and L/λ indicated that the ratio of fluid velocity to transport velocity fitted to the ADE model depends on σb/〈b〉 and L/λ. The continuous time random walk (CTRW) performed much better across all transport scales, and resulted in scale-independent fitted parameters, i.e., β in the memory function. The fitted β is proportional to σb/〈b〉but is insensitive to L/λ. Therefore, bulk longitudinal solute transport across the pre-asymptotic and asymptotic regimes can be estimated based on the CTRW model parameterized by measurable fracture physical properties.

Modeling Bromide Transport in Undisturbed Soil Columns with the Continuous Time Random Walk

Nowadays, mathematical models are commonly used as efficient tools in solute transport studies and management in porous media. In this paper, we evaluated the continuous time random walk (CTRW) theory for its capability to characterize bromide transport in one-dimensional saturated undisturbed clay loam and sandy loam soil columns (10 cm in diameter and 40 cm long). The transport process was also simulated by using the advection–dispersion equation (ADE) for comparison. The values of CTRW parameter, β, for clay loam and sandy loam soil columns were derived at 1.71 and 1.73, respectively. This indicated that the bromide transport behavior within the clay loam and sandy loam soil columns was anomalous transport or non-Fickian transport and CTRW model was more suitable for simulation of bromide transport through undisturbed clay loam and sandy soil columns compared to ADE model with Fickian diffusion law basis. The graphical and statistical analysis confirmed the efficiency of CTRW model for simulation of bromide transport.

Impact of space‐time mesh adaptation on solute transport modeling in porous media

AbstractWe implement a space‐time grid adaptation procedure to efficiently improve the accuracy of numerical simulations of solute transport in porous media in the context of model parameter estimation. We focus on the Advection Dispersion Equation (ADE) for the interpretation of nonreactive transport experiments in laboratory‐scale heterogeneous porous media. When compared to a numerical approximation based on a fixed space‐time discretization, our approach is grounded on a joint automatic selection of the spatial grid and the time step to capture the main (space‐time) system dynamics. Spatial mesh adaptation is driven by an anisotropic recovery‐based error estimator which enables us to properly select the size, shape, and orientation of the mesh elements. Adaptation of the time step is performed through an ad hoc local reconstruction of the temporal derivative of the solution via a recovery‐based approach. The impact of the proposed adaptation strategy on the ability to provide reliable estimates of the key parameters of an ADE model is assessed on the basis of experimental solute breakthrough data measured following tracer injection in a nonuniform porous system. Model calibration is performed in a Maximum Likelihood (ML) framework upon relying on the representation of the ADE solution through a generalized Polynomial Chaos Expansion (gPCE). Our results show that the proposed anisotropic space‐time grid adaptation leads to ML parameter estimates and to model results of markedly improved quality when compared to classical inversion approaches based on a uniform space‐time discretization.

On the mechanism of field‐scale solute transport: Insights from numerical simulations and field observations

AbstractField‐scale transport of conservative and reactive solutes through a deep vadose zone was analyzed by means of two different model processes for the local description of the transport. The first is the advection‐dispersion equation (ADE) model, and the second is the mobile‐immobile (MIM) model. The analyses were performed by means of three‐dimensional (3‐D), numerical simulations of flow and transport considering realistic features of the flow system, pertinent to a turf field irrigated with treated sewage effluents (TSE). Simulated water content and concentration profiles were compared with available measurements of their counterparts. Results of the analyses suggest that the behavior of both solutes in the deep vadose zone of the Glil Yam site is better quantified by the MIM model than by the ADE model. Reconstruction of the shape of the measured solute concentration profiles using the MIM model required relatively small mass transfer coefficient, γ, and relatively large volume fraction of the immobile water θim. This implies that for an initially nonzero solute concentration profile, as compared with the MIM model, the ADE model may significantly overestimate the rate at which solutes are loaded in the groundwater. On the contrary, for an initially zero solute concentration profile, as compared with the MIM model, the ADE model may significantly underestimate solute velocities and early arrival times to the water table. These findings stem from the combination of relatively small γ and relatively large θim taken into account in the MIM model. In the first case, this combination forces a considerable portion of the solute mass to reside in the immobile region of the water‐filled pore space, while the opposite is true in the second case.

Experimental study of solute transport under non-Darcian flow in a single fracture

Summary We have experimentally studied solute transport in a single fracture (SF) under non-Darcian flow condition which was found to closely follow the Forchheimer equation at Reynolds numbers around 12.2–86.0 when fracture apertures were between 4 mm and 9 mm. The measured breakthrough curves (BTCs) under the non-Darcian flow condition had some features that are difficult to explain using the Fickian type advection–dispersion equation (ADE). All the measured BTCs showed long tails, which might be caused by mass transfer between the boundary layer near the fracture wall and the mobile domain near the plane of symmetry, as supported by the boundary layer dispersion theories of Koch and Brady, 1985 , Koch and Brady, 1987 . A mobile–immobile (MIM) model was used to simulate the measured BTCs. To show that the MIM model was doing a better job than the ADE model in describing the observed BTCs, we conducted statistical analysis on the goodness of fitting with these two models. The results showed that the correlation coefficients for the MIM model were greater than those for the ADE model and were close to unity, indicating a nearly perfect fit with the MIM. The mass transfer rate between the mobile domain and the boundary layer increased when the mobile water fraction became larger. The best fit dispersivity values using the MIM model varied between 1.05 mm and 9.29 mm whereas their counterparts using the ADE model varied between 245 mm and 462 mm for the experimental condition of this study. Several issues such as the possible bimodal concentration distribution and the scale-dependent transport in a SF were discussed and would be investigated in the future.